Linnik问题的方差估计
Variance estimates in Linnik's problem
摘要：
我们评估了在一个以原点为中心的三维球面表面上的一个随机旋转的小球中的格点数目的方差。在此之前，Bourgain、Rudnick和Sarnak在广义Lindelöf假设的基础上证明了方差由$σ(Ω_n){N_n}^{1+\varepsilon}$约束，其中$σ(Ω_n)$是单位球面上球的面积$Ω_n$，$N_n$是丢番图方程$x^2 + y^2 + z^2 = n$的解的总数。在大黎曼假设下，利用Soundararajan和Harper的矩方法，建立了$cσ(Ω_n) N_n$形式的上界，其中$c$是一个绝对常数.这个界限是严格的数量级。
英文摘要：
We evaluate the variance of the number of lattice points in a small randomly rotated spherical ball on a surface of 3-dimensional sphere centered at the origin. Previously, Bourgain, Rudnick, and Sarnak showed conditionally on the Generalized Lindelöf Hypothesis that the variance is bounded from above by $σ(Ω_n){N_n}^{1+\varepsilon}$, where $σ(Ω_n)$ is the area of the ball $Ω_n$ on the unit sphere, $N_n$ is the total number of solutions of Diophantine equation $x^2 + y^2 + z^2 = n$. Assuming the Grand Riemann Hypothesis and using the moments method of Soundararajan and Harper, we establish the upper bound of the form $cσ(Ω_n) N_n$, where $c$ is an absolute constant. This bound is of the conjectured order of magnitude.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
